Symmetric bi-skew maps and symmetrized motion planning in projective spaces

TL;DR

This paper explores the differences between two symmetric topological complexities of projective spaces, characterizes them via symmetric maps, and computes these complexities for certain real projective spaces, advancing understanding of their embedding dimensions.

Contribution

It characterizes symmetric topological complexities of real projective spaces through symmetric maps and computes these complexities for specific cases, linking to classical embedding problems.

Findings

$TC^ ext{Sigma}(RP^m)$ characterized by symmetric $bZ_2$-biequivariant maps

Computed $TC$ numbers for $RP^{2^e}$, $e eq 0$

Identified the torus $S^1 imes S^1$ as the only unknown case

Abstract

This work is motivated by the question of whether there are spaces $X$ for which the Farber-Grant symmetric topological complexity $TC^S(X)$ differs from the Basabe-Gonz\'alez-Rudyak-Tamaki symmetric topological complexity $TC^\Sigma(X)$. It is known that, for a projective space $RP^m$, $TC^S(RP^m)$ captures, with a few potentially exceptional cases, the Euclidean embedding dimension of $RP^m$. We now show that, for all $m\geq1$, $TC^\Sigma(RP^m)$ is characterized as the smallest positive integer $n$ for which there is a symmetric $\mathbb{Z}_2$-biequivariant map $S^m\times S^m\to S^n$ with a "monoidal" behavior on the diagonal. This result thus lies at the core of the efforts in the 1970's to characterize the embedding dimension of real projective spaces in terms of the existence of symmetric axial maps. Together with Nakaoka's description of the cohomology ring of symmetric squares, this allows us to compute both $TC$ numbers in the case of $RP^{2^e}$ for $e\geq1$. In particular, this leaves the torus $S^1\times S^1$ as the only closed surface whose symmetric (symmetrized) $TC^S$ ($TC^\Sigma$) -invariant is currently unknown.